Infi-Supermartingale Property

• The infi-supermartingale property is defined as a process meeting an infimum constraint over conditional expectations across a family of measures, ensuring minimal drift under uncertainty.
• It plays a key role in robust asset pricing and supermartingale optimal transport by calibrating processes to an infinite collection of marginal constraints and delineating worst-case scenarios.
• Distinct from classical and G-supermartingale notions, this property captures the minimal necessary drift control, facilitating practical strategies in stochastic control and robust finance.

The Infi-Supermartingale property characterizes stochastic processes under constraints involving infinitely many marginals or model uncertainty, where comparison is made via an infimum or continuum of conditional expectations rather than the classical filtration of a single probability law. The property arises prominently in robust mathematical finance (e.g., asset price bubbles under model uncertainty and short sales prohibition), supermartingale optimal transport with full marginal constraints, and the study of stochastic control problems on infinite horizons. This structural feature sits between the absence of restriction and the uniform “worst-case” or “best-case” supermartingale property, making it a fundamental concept in modern stochastic analysis.

1. Formal Definitions and Basic Framework

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $(\mathcal{F}_t)_{t\geq0}$ a filtration, and consider a family $\mathcal{Q}$ of probability measures, possibly non-dominated. An adapted process $(Y_t)_{0 \leq t \leq T}$ is called an infi-supermartingale (relative to $\mathcal{Q}$) if

$$Y_t \geq \inf_{Q \in \mathcal{Q}} E_Q[ Y_T \mid \mathcal{F}_t ] \quad \forall\, 0 \leq t \leq T.$$

This is equivalent to non-negativity of “drift” with respect to the lower nonlinear (infimum) conditional expectation,

$Y_t \geq \ell \mathbb{E}[Y_T | \mathcal{F}_t],$

where $$\ell \mathbb{E}[\cdot|\mathcal{F}_t] := \inf_{Q \in \mathcal{Q}} E_Q[\cdot|\mathcal{F}_t]$$ (Zhang, 24 Dec 2025).

In continuous-time supermartingale optimal transport with full-marginals constraint, the term infi-supermartingale refers to processes calibrated to an uncountable collection of marginal laws $(\mu_t)_{t \in [0,1]}$, for instance, on $D([0,1],\mathbb{R})$, where the set $\mathcal{P}^\infty(\mu)$ consists of supermartingale measures matching the full marginal specification (Bayraktar et al., 2022).

2. Infi-Supermartingale Property in Robust Asset Pricing and Bubbles

In discrete-time asset pricing models under model uncertainty and short-sales prohibitions, the infi-supermartingale property plays a pivotal role in the characterization of price bubbles and their persistence:

• Given a process $(S_t)$ and the family $\mathcal{Q}$ of risk-neutral measures, the bubble process $\beta_t := S_t - S^*_t$ (with $S^*_t$ as the “fundamental price”) must satisfy, in the case of unbounded maturity $\tau$,

$$\beta_t \geq \inf_{Q \in \mathcal{Q}} E_Q[\beta_T | \mathcal{F}_t],$$

i.e., $(\beta_t)$ is an infi-supermartingale (Zhang, 24 Dec 2025).

• If the horizon is finite ($\tau\leq T < \infty$), a stronger property holds: the $G$-supermartingale property, corresponding to the supremum over $\mathcal{Q}$.
• Economically, this reflects that the bubble cannot fall below the smallest conditional expectation taken over all plausible (possibly adversarial) models; it encapsulates persistence of bubble value in the most pessimistic scenario, sharply distinguishing it from bounds using only the classical supermartingale (single-prior) or G-supermartingale (supremum) notions.
• The necessity of the infi-supermartingale property in the case of unbounded maturity arises because, unlike the finite-horizon case, the supremum condition is not enforced by the definition of a bubble—it is only the minimal (infimum) drift that is necessarily present (Zhang, 24 Dec 2025).

Maturity Structure	Minimal Necessary Property	Sufficient Property
Unbounded ($\tau$)	Infi-supermartingale	$G$-supermartingale
Bounded ($\tau\leq T$)	$G$-supermartingale	$G$-supermartingale

3. Infi-Supermartingale in Supermartingale Optimal Transport with Full Marginals

In the context of continuous-time supermartingale optimal transport (SMOT), the infi-supermartingale property is realized as the requirement that the process matches a continuum of marginal measures and transports them under a supermartingale coupling:

• For target flows of laws $(\mu_t)_{t \in [0,1]}$, the admissible set $\mathcal{P}^\infty(\mu)$ contains all processes with the prescribed marginals and supermartingale property.
• The optimal transport problem is to maximize $E^P[C(X_\cdot)]$ over $P \in \mathcal{P}^\infty(\mu)$, where $C$ is a path-dependent cost; this requires a calibration to all marginals—a canonical infinite-horizon/infi-marginal construction (Bayraktar et al., 2022).
• The construction uses Markovian iteration of one-period optimal supermartingale couplings, passing to the limit over increasingly fine partitions, and characterizes the optimizer as a pure-jump Markov process of local-Lévy type, precisely calibrated to the continuum of marginals, embodying the infi-supermartingale property.
• The necessity of the infi-supermartingale constraint here arises from Strassen–Yor–Ewald conditions: the continuum of convex-decreasing orderings and right-continuity of the marginals are both required for the existence of an admissible process (Bayraktar et al., 2022).

4. Relation to Classical Supermartingale and $G$-Supermartingale Properties

The infi-supermartingale property generalizes the single-prior supermartingale by considering an infimum over a set of plausible measures:

• Classical supermartingale: $E_Q[Y_{t+1} | \mathcal{F}_t] \leq Y_t$ for one fixed measure $Q$.
• $G$-supermartingale: $$Y_t \geq \sup_{Q \in \mathcal{Q}} E_Q[Y_T | \mathcal{F}_t]$$.
• Infi-supermartingale: $$Y_t \geq \inf_{Q \in \mathcal{Q}} E_Q[Y_T | \mathcal{F}_t]$$.

When $\mathcal{Q}$ is a singleton, all three notions collapse. For finite horizons and uniformly integrable cash-flows, $\inf = \sup$, and distinctions vanish. In infinite- or unbounded-maturity regimes, the infimum and supremum can diverge, with the infimum yielding only minimal (but necessary) drift control (Zhang, 24 Dec 2025).

5. Illustrative Examples and Explicit Constructions

Both the robust bubble literature and SMOT provide explicit constructions showing the necessity and operation of the infi-supermartingale property:

• Fiat money under unbounded maturity (asset with no dividends, infinite horizon, price process defined by inverse monetary base): the entire price process is a nontrivial infi-supermartingale but not necessarily a $G$-supermartingale. Minimum conditional expectations strictly control possible downward movements (Zhang, 24 Dec 2025).
• Supermartingale OT with uniform, normal (Bachelier), and geometric Brownian marginals: Convergence of iterated one-step couplings in the Markovian framework yields a limiting process—a pure-jump Markov process with explicit drift and intensity—calibrated to all marginals and solving the path-dependent optimal transport problem (Bayraktar et al., 2022).

6. Methodological and Economic Implications

The infi-supermartingale property is crucial in several respects:

• Model Uncertainty/Robustness: Ensures process constraints are satisfied under the most pessimistic/favorable evolution consistent with all plausible models.
• Minimal-drift Constraints: For bubbles or prices, the property enforces that observed value cannot fall below the worst-case future expectation.
• Optimal Transport and Calibration: For processes spanned by an infinite marginal family, the property guarantees calibration while maintaining supermartingale structure.
• Limiting Regimes and Duality: Facilitates the proof of strong duality and construction of optimal strategies in robust stochastic control and transport problems.

7. Connections to Related Properties and Theorems

• Optional-Stopping and Superhedging: The infi-supermartingale property justifies use of stopping-time arguments and superhedging duality in robust or infinite-marginal settings.
• Submartingale/Supermartingale Regimes: The boundary between infi-supermartingale and stronger supermartingale properties is sharp; in stochastic games or optimization (e.g., the Kelly criterion), the infinite-horizon supermartingale regime is characterized by critical parameter thresholds, with direct economic interpretations (e.g., certain ruin beyond the zero-growth threshold) (Miller, 24 Feb 2025).
• Dual Representations and Nonlinear Expectations: The interplay of upper/lower nonlinear expectations (sup/inf over measure families) is central to the modern robust analysis of uncertainty, pricing, and calibration.

In summary, the infi-supermartingale property defines a necessary, minimal-drift stochastic process constraint tailored to infinite-horizon, full-marginals, or robust model settings. It ensures process consistency under adverse modeling assumptions or dense marginal constraints, and features prominently in contemporary research in robust finance and optimal transport (Bayraktar et al., 2022, Zhang, 24 Dec 2025).